Theorem hoscli

Description: Closure of Hilbert space operator sum. (Contributed by NM, 12-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hoeq.1	⊢ 𝑆 : ℋ ⟶ ℋ
	hoeq.2	⊢ 𝑇 : ℋ ⟶ ℋ
Assertion	hoscli	⊢ ( 𝐴 ∈ ℋ → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )

Proof

Step	Hyp	Ref	Expression
1		hoeq.1	⊢ 𝑆 : ℋ ⟶ ℋ
2		hoeq.2	⊢ 𝑇 : ℋ ⟶ ℋ
3		hoscl	⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )
4	1 2 3	mpanl12	⊢ ( 𝐴 ∈ ℋ → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )